A002468 -id:A002468 - cp's OEIS frontend

A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, 76894368849186894, 1537887376983737879, 32295634916658495460, 710503968166486900119

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the number of permutations in the symmetric group S_n that have a fixed point, i.e., they are not derangements (A000166). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001
a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=k! for k=0,1,...,n. - Michael Somos, Oct 07 2003
The termwise sum of this sequence and A000166 gives the factorial numbers. - D. G. Rogers, Aug 26 2006, Jan 06 2008
a(n) is the number of deco polyominoes of height n and having in the last column an odd number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the horizontal domino is the only deco polyomino of height 2 having an odd number of cells in the last column. - Emeric Deutsch, May 08 2008
Starting (1, 4, 15, 76, 455, ...) = eigensequence of triangle A127899 (unsigned). - Gary W. Adamson, Dec 29 2008
(n-1) | a(n), hence a(n) is never prime. - Jonathan Vos Post, Mar 25 2009
a(n) is the number of permutations of [n] that have at least one fixed point = number of positive terms in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012
Numerator of partial sum of alternating harmonic series, provided that the denominator is n!. - Richard Locke Peterson, May 11 2020
a(n) is the number of terms in the polynomial expansion of the determinant of a n X n matrix that contains at least one diagonal element. - Adam Wang, May 28 2025

			G.f. = x + x^2 + 4*x^3 + 15*x^4 + 76*x^5 + 455*x^6 + 3186*x^7 + 25487*x^8 + ...

R. K. Guy, Unsolved Problems Number Theory, E37.
R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
Sergi Elizalde, Bijections for restricted inversion sequences and permutations with fixed points, arXiv:2006.13842 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, Feb 10 1993
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
R. K. Guy and S. Washburn, Correspondence, Nov. - Dec. 1991
T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.
J. Metzger, Email to N. J. A. Sloane, Apr 30 1991
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Exact formulas for Integer Sequences.
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraphs 5 and 7.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002468, A002469, A028306, A047920, A052169, A127899, A276975.
Row sums of A068106.
Column k=1 of A293211.
Column k=0 of A299789, A306234, and of A324362.

a := proc(n) -add((-1)^i*binomial(n, i)*(n-i)!, i=1..n) end;
a := n->-n!*add((-1)^k/k!, k=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, May 25 2007
a := n -> simplify(GAMMA(n+1) - GAMMA(n+1, -1)*exp(-1)):
seq(a(n), n=0..20); # Peter Luschny, Feb 28 2017

Denominator[k=1; NestList[1+1/(k++ #1)&,1,12]] (* Wouter Meeussen, Mar 24 2007 *)
a[ n_] := If[ n < 0, 0, n! - Subfactorial[n]] (* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 1, 0, n! - Round[ n! / E]] (* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 0, 0, n! - (-1)^n HypergeometricPFQ[ {- n, 1}, {}, 1]](* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - Exp[ -x] ) / (1 - x), {x, 0, n}]] (* Michael Somos, Jan 25 2014 *)
RecurrenceTable[{a[n] == (n - 1) ( a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *)

{a(n) = if( n<1, 0, n * a(n-1) - (-1)^n)} /* Michael Somos, Mar 24 2003 */

{a(n) = if( n<0, 0, n! * polcoeff( (1 - exp( -x + x * O(x^n))) / (1 - x), n))} /* Michael Somos, Mar 24 2003 */

a(n) = if(n<1,0,subst(polinterpolate(vector(n,k,(k-1)!)),x,n+1))

A002467(n) = if(n<1, 0, n*A002467(n-1)-(-1)^n); \\ Joerg Arndt, Apr 22 2013

a(n) = n! - A000166(n) = A000142(n) - A000166(n).
E.g.f.: (1 - exp(-x)) / (1 - x). - Michael Somos, Aug 11 1999
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1; a(1) = 1. - Michael Somos, Aug 11 1999
a(n) = n*a(n-1) - (-1)^n. - Michael Somos, Aug 11 1999
a(0) = 0, a(n) = floor(n!(e-1)/e + 1/2) for n > 0. - Michael Somos, Aug 11 1999
a(n) = - n! * Sum_{i=1..n} (-1)^i/i!. Limit_{n->infinity} a(n)/n! = 1 - 1/e. - Gerald McGarvey, Jun 08 2004
Inverse binomial transform of A002627. - Ross La Haye, Sep 21 2004
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1. - Gary Detlefs, Apr 11 2010
a(n) = n! - floor((n!+1)/e), n > 0. - Gary Detlefs, Apr 11 2010
For n > 0, a(n) = {(1-1/exp(1))*n!}, where {x} is the nearest integer. - Simon Plouffe, conjectured March 1993, added Feb 17 2011
0 = a(n) * (a(n+1) + a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2) * (a(n+2)) if n >= 0. - Michael Somos, Jan 25 2014
a(n) = Gamma(n+1) - Gamma(n+1, -1)*exp(-1). - Peter Luschny, Feb 28 2017
a(n) = Sum_{k=0..n-1} A047920(n-1,k). - Alois P. Heinz, Sep 01 2021

A002469 The game of Mousetrap with n cards: the number of permutations of n cards in which 2 is the only hit.

Original entry on oeis.org

0, 0, 1, 5, 31, 203, 1501, 12449, 114955, 1171799, 13082617, 158860349, 2085208951, 29427878435, 444413828821, 7151855533913, 122190894996451, 2209057440250799, 42133729714051825, 845553296311189109, 17810791160738752207, 392911423093684031099

Offset: 2

N. J. A. Sloane

			G.f.: x^4 + 5*x^5 + 31*x^6 + 203*x^7 + 1501*x^8 + 12449*x^9 + 114955*x^10 + ...

R. K. Guy, Unsolved Problems Number Theory, E37.
R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 2..100
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
J. Metzger, Email to N. J. A. Sloane, Apr 30 1991
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
M. Z. Spivey, Staircase rook polynomials and Cayley's game of mousetrap, Eur. J. Combinat. 30 (2) (2009) 532-539
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002468, A002467, A028306, A159610, A000255, A000166.

A002469:=n->(n-3)*floor(((n-2)!+1)/exp(1)) + (n-4)*floor(((n-3)!+1)/exp(1)): 0, seq(A002469(n), n=3..30); # Wesley Ivan Hurt, Jan 10 2017

Join[{0},Table[(n-3)Floor[((n-2)!+1)/E]+(n-4)Floor[((n-3)!+1)/E], {n,3,30}]] (* Harvey P. Dale, Feb 05 2012 *)
a[n_] := (n-3)*Subfactorial[n-2]+(n-4)*Subfactorial[n-3]; a[n_ /; n <= 3] = 0; Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Dec 12 2014 *)

default(realprecision,200);
e=exp(1);
A002469(n) = if( n<=3, 0, (n-3)*floor(((n-2)!+1)/e) + (n-4)*floor(((n-3)!+1)/e) );
/* Joerg Arndt, Apr 22 2013 */

a(n) = sum of terms in (n-2)-nd row of triangle A159610; equivalent to: a(n) = (n-2)*A000255(n-1) + A000166(n). -  Gary W. Adamson, Apr 17 2009
a(n) = (n-3)* A000166(n-2) + (n-4)* A000166(n-3). - Gary Detlefs, Apr 10 2010
a(n) = (n-3)*floor(((n-2)!+1)/e) + (n-4)*floor(((n-3)!+1)/e), for n>2. - Gary Detlefs, Apr 10 2010
G.f.: x - 1 + (1-2*x)/(x*Q(0)), where Q(k) = 1/x - (2*k+1) - (k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013

More terms from Harvey P. Dale, Feb 05 2012

A018934 From the game of Mousetrap.

Original entry on oeis.org

0, 0, 0, 2, 8, 42, 256, 1810, 14568, 131642, 1320128, 14551074, 174879880, 2276108362, 31894886208, 478775722802, 7664993150696, 130369025763930, 2347604596782208, 44619881467365442, 892659329531868168, 18750556523491299434, 412601744979927877760, 9491630163800726992722

Offset: 0

N. J. A. Sloane

nonn

Number of permutations p of [n] such that p(k) = k+2 for exactly one k in the range 0 < k < n-1. - Vladeta Jovovic, Nov 30 2007

Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.

Cf. A000166, A002468, A055790.

Join[{0,0},With[{nn=30},CoefficientList[Series[(2x Exp[-x])/(1-x)^3, {x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Nov 16 2013 *)

C=binomial;
a(n)=if(n<=2, 0, n! + sum(k=1,n, (-1)^k * ( C(n-1,k)+C(n-2,k-1) )*(n-k)! ) );
/* Joerg Arndt, Apr 22 2013 */

def A():
    a, b, n  = 1, 1, 1
    yield 0
    while True:
        yield b - a
        n += 1
        a, b = b, (n-2)*a+(n-1)*b
A018934 = A()
print([next(A018934) for  in range(24)]) # _Peter Luschny, Jan 30 2017

From Vladeta Jovovic, Nov 30 2007: (Start)
a(n) = (n-2)*A055790(n-2).
E.g.f.: 2*x*exp(-x)/(1-x)^3. (End)
a(n) = floor((n!+1)/e) - floor(((n-2)!+1)/e), n > 2. - Gary Detlefs, Mar 27 2011
G.f.: (1-x)*x/Q(0) - x, where Q(k) = 1 + x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: G(0)*x - x, where G(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - (1-x*(1+2*k))*(1-x*(3+2*k))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 05 2014
For n > 1, a(n) = (n-1)*A000166(n-1) + (n-2)*A000166(n-2). - Kevin Long, Feb 21 2021

More terms from Vladeta Jovovic, Nov 30 2007, corrected Jan 25 2008

A018932 The number of permutations of n cards in which 4 will be the next hit after 2.

Original entry on oeis.org

0, 3, 10, 60, 408, 3120, 26640, 252000, 2620800, 29756160, 366508800, 4869849600, 69455232000, 1058593536000, 17174123366400, 295534407168000, 5377157001216000, 103149354147840000, 2080771454361600000

Offset: 4

N. J. A. Sloane

nonn

From the game of Mousetrap.

G. C. Greubel, Table of n, a(n) for n = 4..450
D. J. Mundfrom, A problem of permutations: the Game of "Mousetrap", Eur. J. Combinat. 15 (1994) 555-560.

Cf. A002468.

Concatenation([0,3], List([6..30], n-> (n^2-8*n+17)*Factorial(n-4) )); # G. C. Greubel, Feb 21 2019

[0,3] cat [(n^2-8*n+17)*Factorial(n-4): n in [6..30]]; // G. C. Greubel, Feb 21 2019

0,3,seq((n^2-8*n+17)*factorial(n-4),n=6..30); # Muniru A Asiru, Feb 22 2019

Join[{0,3}, Table[(n^2-8*n+17)*(n-4)!, {n,6,30}]] (* G. C. Greubel, Feb 21 2019 *)

for(n=4,30, print1(if(n==4, 0, if(n==5, 3, (n^2-8*n+17)*(n-4)!)), ", ")) \\ G. C. Greubel, Feb 21 2019

[0,3] + [(n^2-8*n+17)*factorial(n-4) for n in (6..30)] # G. C. Greubel, Feb 21 2019

a(n) = (n-2)! - 3*(n-3)! + 2*(n-4)! if n > 5. - R. J. Mathar, Oct 02 2008

E.g.f.: (x*(1020 - 1290*x + 340*x^2 - 15*x^3 + 3*x^4) + 60*(17 - 30*x + 15*x^2 - 2*x^3)*log(1-x))/360. - G. C. Greubel, Feb 21 2019

Offset changed to 4, more terms, better definition and link from R. J. Mathar, Oct 02 2008

A018931 The number of permutations of n cards in which 2 is the first card hit and 3 the next hit after 2.

Original entry on oeis.org

0, 1, 2, 12, 72, 480, 3600, 30240, 282240, 2903040, 32659200, 399168000, 5269017600, 74724249600, 1133317785600, 18307441152000, 313841848320000, 5690998849536000, 108840352997376000, 2189611807358976000, 46225138155356160000, 1021818843434188800000

Offset: 3

N. J. A. Sloane

nonn

From the game of Mousetrap.

Daniel J. Mundfrom, A problem in permutations: the game of "Mousetrap". European J. Combin. 15 (1994), no. 6, 555-560.

D. J. Mundfrom, A problem of permutations: the Game of "Mousetrap", Eur. J. Combinat. 15 (1994) 555-560, Table 1.

Cf. A002468.

a(n) = A062119(n-3), n > 4. - R. J. Mathar, Oct 02 2008

Offset changed to 3 and more precise definition provided by R. J. Mathar, Oct 02 2008

A018933 From the game of Mousetrap.

Original entry on oeis.org

2, 11, 50, 348, 2712, 23520, 225360, 2368800, 27135360, 336752640, 4503340800, 64585382400, 989138304000, 16115529830400, 278360283801600, 5081622594048000, 97772197146624000, 1977622100213760000

Offset: 0

N. J. A. Sloane

nonn

Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.

Cf. A002468.

c := proc(n,x) local a,i; if n > x+1 then a := (n-2)! ; for i from 3 to x do a := a+(-1)^i*(binomial(x-2,i-2)+binomial(x-3,i-3))*(n-i)! ; od: fi; a ; end: A018933 := proc(n) if n = 5 then 2 ; elif n = 6 then 11 ; else c(n,5) ; fi: end: for n from 5 to 23 do printf("%d,",A018933(n)) ; od: # R. J. Mathar, Oct 02 2008

c[n_, x_] := Module[{a = 0, i}, If[n > x+1, a = (n-2)!; For[i = 3, i <= x, i++, a += (-1)^i (Binomial[x-2, i-2] + Binomial[x-3, i-3]) (n-i)!]]; a];
b[n_] := If[n == 5, 2, If[n == 6, 11, c[n, 5]]];
a[n_] := b[n + 5];
a /@ Range[0, 17] (* Jean-François Alcover, Apr 05 2020, after R. J. Mathar *)

This entry was corrupted by a misplaced edit Nov 30 2007; previous (and correct) version restored by N. J. A. Sloane Jan 25 2008

More terms from R. J. Mathar, Oct 02 2008

cp's OEIS Frontend

A260783 Erroneous version of A002468.

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A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

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A002469 The game of Mousetrap with n cards: the number of permutations of n cards in which 2 is the only hit.

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A018934 From the game of Mousetrap.

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A018932 The number of permutations of n cards in which 4 will be the next hit after 2.

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A018931 The number of permutations of n cards in which 2 is the first card hit and 3 the next hit after 2.

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A018933 From the game of Mousetrap.

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